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Localization Length of the \(1+1\) Continuum Directed Random Polymer

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Abstract

In this paper, we study the localization length of the \(1+1\) continuum directed polymer, defined as the distance between the endpoints of two paths sampled independently from the quenched polymer measure. We show that the localization length converges in distribution in the thermodynamic limit, and derive an explicit density formula of the limiting distribution. As a consequence, we prove the \(\tfrac{3}{2}\)-power law decay of the density, confirming the physics prediction of Hwa and Fisher (Phys Rev B 49(5):3136, 1994). Our proof uses the recent result of Das and Zhu (Localization of the continuum directed random polymer, 2022).

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Acknowledgements

A.D. was partially supported by the NSF Mathematical Sciences Postdoctoral Research Fellowship Program through grant no. DMS-2002118. Y.G. was partially supported by the NSF through DMS-2203014. We thank the anonymous referee for suggesting some relevant references.

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Authors and Affiliations

  1. Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Alexander Dunlap

  2. Department of Mathematics, University of Maryland, College Park, Md, 20742, USA

    Yu Gu

  3. Department of Mathematics, University of Michigan, Ann Arbor, Mi, 48109, USA

    Liying Li

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  1. Alexander Dunlap
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  2. Yu Gu
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Correspondence to Yu Gu.

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Communicated by Christian Maes.

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Dunlap, A., Gu, Y. & Li, L. Localization Length of the \(1+1\) Continuum Directed Random Polymer. Ann. Henri Poincaré 24, 2537–2555 (2023). https://doi.org/10.1007/s00023-023-01288-z

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